# What can be deduced from the following hypothesis test given the $p$ value?

You want to find out whether a new medicine for flu is effective or not. You know that the general assumption is that an average flu last for about one week. So, letting $$t$$ be the average disease duration, you formulate your null hypothesis as $$H_0: t \geq 7$$ vs. $$H_1: t < 7$$. Your test result returned a $$p$$ value of $$0.05$$.

Which of the following statements are correct?

1. After treatment with the new drug there is a 95% chance that a flu will last less than 7 days.
2. In 95% of the cases, the flu will last less than 7 days.
3. The duration of the flu can be reduced by a factor of 0.05%.
4. Even if the new medicine was completely ineffective, the chances that the experiment would have produced the same result are 0.05%.
5. We have proven that the null-hypothesis H0 is wrong and your medicine is effective.

Since the $$p$$ value is small we will reject the null hypothesis which is on an avg flu lasts more than or equal to 7 days. By that answer 1 is CORRECT and answer 5 as well. What about the rest? How do I decide that?

• Welcome to CV. If this question relates to a class exercise, please see stats.stackexchange.com/tags/self-study/info and add the tag to modify the question accordingly. Commented Dec 16, 2021 at 10:45
• thank you.. i went thru the guidelines. Commented Dec 16, 2021 at 10:58
• IMHO, faulty or confusing statements of question and proposed answers prevent any sensible response.\\ I vote for five W's. \\ If new medicine has NO effect, then the probability of getting a T statistic smaller than or equal to the one observed is $0.05.$ So you (just barely) reject $H_0: \mu = 7$ in favor of $H_a: \mu < 7$ at the 5% level. Commented Dec 16, 2021 at 17:18
• "So, letting $t$ be the average disease duration, you formulate your null hypothesis as $H_0: t \geq 7$ vs. $H_1: t < 7$" Reading in between the lines I can guess what is meant with this hypothesis. But being pedantic... what does $t$ mean? The average disease duration when the medicine is taken? Commented Dec 27, 2021 at 20:32

None of the options provided are true. This might be an example of the fairly common disease of statistics instructors not understanding p-values. Many other examples can be found. See this paper: https://bpspubs.onlinelibrary.wiley.com/doi/full/10.1111/j.1476-5381.2012.01931.x Also Haller & Krause (2002).

For a significance test, what you know from the p-value of 0.05 is that there is pretty weak statistical evidence suggesting that the null hypothesis is false. For a hypothesis test your decision to reject the null or not would depend on whether your pre-defined critical cutoff (rejection region) was greater or less than 0.05. If you used the convention of p<0.05 as the cutoff then your result of p=0.05 would give the opposite decision from what you would get if you set the cutoff to p≤0.05!

Haller H, Krauss S (2002). Misinterpretations of significance: a problem students share with their teachers. Methods Psychol Res 7: 1–20.

As @MichaelLew said,none of the answers is 100% true. The question and answers are poorly worded.

• Answer 1 is about the distribution of flu duration among people. The study is only about average length and you have no information about the distribution among different people.

• Answer 2 is also about the distribution of flu duration among people, just asking it differently. Nothing in the problem gives a clue about the distribution.

• Answer 3 is about relative duration of flu. Nothing in the problem gives you information about the durations or the relative duration.

• Answer 4 is wrong because of "same result". P value is about same result or even more extreme result (or same or more extreme difference in the other direction, if two-sided p-value). Moreover, answer 4 incorrectly equates a probability of 0.05 with 0.05% (rather than 5%).

• Answer 5 says "proven". Statistics can never absolutely prove in one study. Statistics is about probabilities.

As written therefore, all answers are wrong. But #4 the closest to being correct.

• All 5 statements are so clearly misinterpretations of test outcomes, as others have said. On top of this, 3 and 4 include 0.05% when presumably they mean 5%. Commented Dec 25, 2021 at 20:03
• @GrahamBornholt. Good point that I had missed. I incorporated it into my answer. Commented Dec 27, 2021 at 18:49
• Another problem with answer 4, which I guess is supposed to be the correct answer, is that it speaks about the 'drug being (in)effective', but the effectiveness is not well defined. The hypothesis test is about the duration of the disease (probably when taking the drug). But with that interpretation of the test there is no proper control group. If the disease duration after taking the medicine is below 7 days then this does not need to be because the drug is effective, it can also be because the general assumption is false (in addition, that assumption 'about one week' is very imprecise). Commented Dec 27, 2021 at 20:45